150 lines
4.9 KiB
Plaintext
150 lines
4.9 KiB
Plaintext
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Complements of NAC-Jack BBS from USENET
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From the Journal of Irreproducible Results, Vol. 35, No. 1, without
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permission. In exchange for that, you can subscribe by writing to them
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at Blackwell Scientific Publications, Inc., Three Cambridge Center,
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Cambridge, MA 02142, USA. Generally mildly humorous to outright silly.
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(No affiliation, just a 20 year happy subscriber).
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Note that since I cannot show overstricken or underlined characters, I
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have taken to placing carats under the digits in question. Any errors are
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almost certainly mine, and not the author's.
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Simplified Mathematics
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Ben Ruekberg
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Cranston, Rhode Island
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As John Allen Paulos points out in his book _Innumeracy, mathematical
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illiteracy and its consequences_, "[some of the blame [for the inability
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of students to do math, aside from `extreme intellectual lethargy' (p.89)]
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...must ultimately lie with teachers who aren't sufficiently capable...
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[themselves]."(p.75) He suggests that others, who are capable in math-
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ematics, help instruct. Toward this noble end, I submit a convenient
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and useful shortcut in dealing with fractions which are the bane of young
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students.
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While teachers often instruct students in simplifying fractions, they
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usually restrict their instruction in this method to cancelling zeroes.
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An example of this is:
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10/100 = 10/100 = 1/10
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^ ^
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What they fail to point out is the much more useful expedient of
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cancelling non-zero digits or groups of digits that lie adjacent to the
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slash. Since this method may be novel to many readers and because of
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possible confusion between the slash indicating "division" and the slash
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indicating "crossing out," (and also because it is a great way to save
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space) the cancelled digits will be underlined.
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For the first example, consider the fraction 16/64. A "6" lies on
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each side of the slash, and, thus, can be cancelled.
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16/64 = 1/4
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^ ^
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Another example of one digit simplification of fractions is 19/95.
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19/95 = 1/5
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^ ^
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The simplification can also be done with two digits which lie in the
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same order on either side of the slash. In 133/3325, the 33's cancel.
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133/3325 = 1/25
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^^ ^^
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Similarly, in 181/8145, the 81's on either side of the slash cancel.
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181/8145 = 1/45
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^^ ^^
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The method works well with larger cancellations also, as illustrated
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by a few random examples.
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30405/40540 = 30/40
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^^^ ^^^
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(by conventional cancellation, 3/4)
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1501/501334 = 1/334
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^^^ ^^^
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47641/641633 = 47/633
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^^^ ^^^
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467956/956955 = 467/955
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^^^ ^^^
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32727/272725 = 3/25
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^^^^ ^^^^
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129612/961289 = 12/89
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^^^^ ^^^^
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277777/7777756 = 2/56 = 1/28
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^^^^^ ^^^^^
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4848484/84848470 = 4/70 = 2/35
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^^^^^^ ^^^^^^
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7407407/407407385 = 7/385
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^^^^^^ ^^^^^^
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49999999999/99999999998 = 4/8 = 1/2
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^^^^^^^^^^ ^^^^^^^^^^
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This method works in simplifying improper fractions as well.
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493991/99199 = 493/99
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^^^ ^^^
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4324/3243 = 4/3
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^^^ ^^^
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56504/5045 = 56/5
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^^^ ^^^
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67402/4024 = 67/4
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^^^ ^^^
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23828/8288 = 23/8
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^^^ ^^^
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At this point the ease of manipulation of this method for simplifying
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otherwise daunting fractions must be obvious. But the skeptical reader
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might believe that I have carefully picked cases that work, rather than
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picking numbers at random. I will demonstrate with the cancellation of
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two consecutive numbers and multiple examples using the second, all with
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three-digit numbers cancelling; then with two fractions which simplify to
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the same fraction, with four-digit numbers cancelling.
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702 (followed by 7):
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182702/7027 = 182/7
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^^^ ^^^
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703 (followed by 7):
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133703/7037 = 133/7
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^^^ ^^^
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703 (followed by 57):
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3703/70357 = 3/57
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^^^ ^^^
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Two fractions which simplify to the same value:
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123762/37623648 = 12/3648 = 1/304
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^^^^ ^^^^
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103135/31353040 = 10/3040 = 1/304
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^^^^ ^^^^
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Only when our school systems adapt innovative approaches to handling
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mathematics, such as this, can America hope to continue to compete in
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today's world. With students in other countries showing superior
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mathematical ability, we cannot afford to delay much longer.
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